Convergence of Bi-Virus Epidemic Models with Non-Linear Rates on Networks -- A Monotone Dynamical Systems Approach

TL;DR

This paper analyzes the convergence of a bi-virus epidemic model with non-linear infection and recovery rates on networks, using monotone dynamical systems theory to classify long-term outcomes.

Contribution

It provides the first complete convergence analysis for nonlinear bi-virus models on general graphs, addressing previous gaps in understanding coexistence scenarios.

Findings

Identifies conditions for virus-free, single-virus, and coexistence states.

Uses monotone dynamical systems to establish global convergence results.

Addresses limitations of Lyapunov-based methods in this context.

Abstract

We study convergence properties of competing epidemic models of the Susceptible-Infected-Susceptible (SIS) type. The SIS epidemic model has seen widespread popularity in modelling the spreading dynamics of contagions such as viruses, infectious diseases, or even rumors/opinions over contact networks (graphs).We analyze the case of two such viruses spreading on overlaid graphs, with non-linear rates of infection spread and recovery. We call this the non-linear bi-virus model and, building upon recent results, obtain precise conditions for global convergence of the solutions to a trichotomy of possible outcomes: a virus-free state, a single-virus state, and to a coexistence state. Our techniques are based on the theory of monotone dynamical systems (MDS), in contrast to Lyapunov based techniques that have only seen partial success in determining convergence properties in the setting of competing epidemics. We demonstrate how the existing works have been unsuccessful in characterizing a large subset of the model parameter space for bi-virus epidemics, including all scenarios leading to coexistence of the epidemics. To the best of our knowledge, our results are the first in providing complete convergence analysis for the bi-virus system with nonlinear infection and recovery rates on general graphs.